= Solution
The force is <central force>, so its <torque> about the star vanishes. The <specific angular momentum> is therefore constant:
$$
h=|\mathbf r\mathbin\times\mathbf v|=r^2\dot\theta.
$$
Taking the scalar product of $\ddot{\mathbf r}=-\mu\mathbf r/r^3$ with $\dot{\mathbf r}$ gives
$$
\frac d{dt}\left(\frac{v^2}{2}-\frac\mu r\right)
=\dot{\mathbf r}\mathbin\cdot\ddot{\mathbf r}
+\frac\mu{r^3}\mathbf r\mathbin\cdot\dot{\mathbf r}=0.
$$
Thus <conservation of energy> gives the second constant
$$
\boxed{C=\frac{v^2}{2}-\frac\mu r}.
$$
Solved by gpt-5.6-sol high.
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